sensors
Article
Analysis of a Segmented Annular Coplanar
Capacitive Tilt Sensor with Increased Sensitivity
Jiahao Guo, Pengcheng Hu * and Jiubin Tan
Received: 11 December 2015; Accepted: 19 January 2016; Published: 21 January 2016
Academic Editor: Vamsy P. Chodavarapu
Department of Automation Measurement and Control Engineering, Harbin Institute of Technology,
D-403 Science Park, 2 Yikuang Street, Harbin 150080, China; [email protected] (J.G.); [email protected] (J.T.)
* Correspondence: [email protected]; Tel.: +86-451-8641-2041 (ext. 803); Fax: +86-451-8640-2258
Abstract: An investigation of a segmented annular coplanar capacitor is presented. We focus on
its theoretical model, and a mathematical expression of the capacitance value is derived by solving
a Laplace equation with Hankel transform. The finite element method is employed to verify the
analytical result. Different control parameters are discussed, and each contribution to the capacitance
value of the capacitor is obtained. On this basis, we analyze and optimize the structure parameters of
a segmented coplanar capacitive tilt sensor, and three models with different positions of the electrode
gap are fabricated and tested. The experimental result shows that the model (whose electrode-gap
position is 10 mm from the electrode center) realizes a high sensitivity: 0.129 pF/˝ with a non-linearity
of <0.4% FS (full scale of ˘40˝ ). This finding offers plenty of opportunities for various measurement
requirements in addition to achieving an optimized structure in practical design.
Keywords: segmented annular coplanar capacitor;
increased sensitivity
structure optimization;
tilt sensing;
1. Introduction
Motivated by the fringing effect existing in conventional capacitors, different types of coplanar
capacitive sensors have been proposed in recent years [1–5]. With the demand for lab-on-a-chip devices
and the need for sensor miniaturization, coplanar capacitive sensors with interdigital electrodes [6–10]
are proposed as one of the most used periodic electrodes configuration. Because of the unique structure
in which the sensor electrodes lie in the same plane, specimens can be easily sensed or tested from
one side of the sensor, instead of within the space between electrodes, which largely expands the
application fields of capacitive sensors. By employing advanced manufacturing techniques, such
coplanar electrodes can be fabricated very tightly, and a relatively high capacitance value can be
easily and stably obtained compared with conventional methods. All these benefits make the coplanar
capacitive sensor a popular option for applications in detecting food quality [1], water intrusion [2,3],
relative humidity [4], and particulate matter [5].
In some particular situations, several works have been completed to design and characterize
coplanar capacitive sensors to meet different measurement requirements [11] where the conventional
rectangular electrodes are replaced with concentric annular ones [12,13]. Such annular coplanar
capacitive sensors are superior in terms of rotational symmetry, and they also possess larger sensing
zones. Accordingly, studies on their mathematical models have been conducted [14–16].
By setting the capacitor in cylindrical coordinates, Chen derived an electrostatic Green’s function
from point charges using the Hankel transform method [14]. By dividing the electrodes into circular
filaments and accumulating the respective charge distribution, the capacitance value was then
calculated. Experiments demonstrated the capability for detecting water intrusion in radome structures.
Sensors 2016, 16, 133; doi:10.3390/s16010133
www.mdpi.com/journal/sensors
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In [15],
a simple closed-form solution for concentric coplanar capacitors was introduced by
Cheng,
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where a Laplace equation on the electrical potential was solved by replacing a Dirichlet boundary
structures.
[15], a simple
closed-form
solution for
concentric
coplanar
introduced
condition
with In
a Neumann
one.
A double-layered
medium
model
of thecapacitors
capacitorwas
was
developed to
by Cheng,
where acorneum
Laplace equation
the electrical
was
solved
by replacing
a Dirichlet
simulate
the stratum
and deepontissue
layer ofpotential
the body.
Such
concentric
coplanar
capacitors
condition
with hydration
a Neumannsensing.
one. A double-layered medium model of the capacitor was
couldboundary
be used for
epidermal
developed to simulate the stratum corneum and deep tissue layer of the body. Such concentric
However, the research work mentioned above focused on an existing model of the annular
coplanar capacitors could be used for epidermal hydration sensing.
capacitor, which consists of an inner central disk and an outer annulus. Further, their applications
However, the research work mentioned above focused on an existing model of the annular
are limited
within
theconsists
material
characteristics,
instead
measurement.
capacitor,
which
of an
inner central disk
andof
angeometrical-dimension
outer annulus. Further, their
applicationsIn the
present
paper,
an
analytical
model
of
a
segmented
annular
coplanar
capacitor
is
proposed,
which
are limited within the material characteristics, instead of geometrical-dimension measurement. In
linksthe
thepresent
centralpaper,
angle an
(a geometrical
dimension)
to
the
capacitance
value
of
a
capacitor.
We
derive
analytical model of a segmented annular coplanar capacitor is proposed,
a mathematical
of the
by solvingtoa the
Laplace
equation
with
transform.
which links expression
the central angle
(a capacitance
geometrical dimension)
capacitance
value
of aHankel
capacitor.
We
derive
a mathematical
expression
capacitance
bytosolving
a Laplace
equation
with On
Hankel
A finite
element
model of the
capacitorofisthe
built
and solved
validate
this analytical
result.
the basis
A finitethe
element
model
of the capacitor
built and
solved
validate this
analytical
of thetransform.
analysis result,
structure
parameters
of a tiltissensor
with
such to
segmented
annular
coplanar
result.
On
the
basis
of
the
analysis
result,
the
structure
parameters
of
a
tilt
sensor
with
such
capacitors are optimized, and a corresponding sensitivity experiment demonstrates the feasibility and
segmented annular coplanar capacitors are optimized, and a corresponding sensitivity experiment
validity of the proposed analytical method.
demonstrates the feasibility and validity of the proposed analytical method.
Compared with conventional capacitive tilt sensors [17–19] using parallel electrodes shown in
Compared with conventional capacitive tilt sensors [17–19] using parallel electrodes shown in
Figure
1a,b,1a,b,
which
suffer
fromfrom
a large
viscous
drag
laglag
and
nonuniformity
between two
Figure
which
suffer
a large
viscous
drag
and
nonuniformityof
ofthe
the distance
distance between
parallel
electrodes,
the
proposed
capacitive
tilt
sensors
[20]
using
annular
coplanar
electrodes
shown
two parallel electrodes, the proposed capacitive tilt sensors [20] using annular coplanar electrodes
in Figure
1c,d
are free
from
problems
and own
anown
excellent
performance.
shown
in Figure
1c,d
are these
free from
these problems
and
an excellent
performance.
Figure 1. Schematic views of capacitive tilt sensors using parallel electrodes: (a) front view and (b)
Figure
1. Schematic views of capacitive tilt sensors using parallel electrodes: (a) front view and (b) side
side view; schematic views of capacitive tilt sensors using annular coplanar electrodes: (c) front view
view; schematic views of capacitive tilt sensors using annular coplanar electrodes: (c) front view and
and (d) side view.
(d) side view.
2. Analytical Model
2. Analytical Model
Figure 2 shows that a segmented annular coplanar capacitor consists of two concentric
electrodes
with central
angle θ0. rii and
rio arecoplanar
inner and
outer radii
of the of
inner
electrode,
Figure
2 shows
that a segmented
annular
capacitor
consists
twoannular
concentric
electrodes
and
roorare
inner
and
outer
radii
of
the
outer
annular
electrode,
respectively.
The
with respectively,
central angle
θ 0 .roiriiand
and
are
inner
and
outer
radii
of
the
inner
annular
electrode,
respectively,
io
coplanar
capacitance
consists
of
three
parts:
two
fringing
capacitances
on
two
sides
of
electrodes
and
and roi and roo are inner and outer radii of the outer annular electrode, respectively. The coplanar
one normal capacitance between electrodes. In a coplanar capacitive sensor, we make use of the
capacitance
consists of three parts: two fringing capacitances on two sides of electrodes and one normal
fringing effect to measure other physical quantities, rather than normal capacitance. Consequently,
capacitance between electrodes. In a coplanar capacitive sensor, we make use of the fringing effect
the electrode thickness should be very thin and neglectable compared with other dimensions [2].
to measure
other physical quantities, rather than normal capacitance. Consequently, the electrode
Due to the symmetry, the electric-field distributions on two sides of electrodes are similar, and we
thickness should be very thin and neglectable compared with other dimensions [2]. Due to the
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symmetry, the electric-field distributions on two sides of electrodes are similar, and we concentrate on
an analytical
model
of analytical
the coplanar
capacitor
with a medium
one aside
for convenience.
concentrate
on an
model
of the coplanar
capacitoronwith
medium
on one side Assume
for
that the
permittivity
and thickness
of the medium
are ε andofhthe
separately.
Potential
the medium
convenience.
Assume
that the permittivity
and thickness
medium are
ε and φ
h in
separately.
satisfies
the Laplace
∆φ = 0.the Laplace equation Δφ = 0.
Potential
φ in theequation
medium satisfies
Figure
2. Schematic
views
a segmentedconcentric
concentric annular
capacitor:
(a) top
view,view;
and and
Figure
2. Schematic
views
of of
a segmented
annularcoplanar
coplanar
capacitor:
(a) top
(b)
section
view.
(b) section view.
When the origin of a cylindrical-coordinate system is set at the center of the electrodes, the
When
origin is
ofexpressed
a cylindrical-coordinate
system is set at the center of the electrodes, the Laplace
Laplacethe
equation
as
equation is expressed as
ˆ
˙
2
2
1 B
Bϕ
11 B 2ϕ
 ` B2ϕ “ 0
∆ϕ
“= 1  rr  `


(1) (1)
2
2

 r 2 Bθ 2 Bz22  0
rrBr
r 
r Br
r
z


where r is the radial distance, θ is the azimuth, and z is the depth. Because the electrodes are swept
r is the radial
distance, azimuth
θ is the azimuth,
z is the depth.factor
Because
electrodes
are swept
alongwhere
the peripheral
direction,
θ is notand
a determinant
ofthe
electric
potential
φ in the
along
the
peripheral
direction,
azimuth
θ
is
not
a
determinant
factor
of
electric
potential
φ in the
medium, and Equation (1) could be rewritten as
medium, and Equation (1) could be rewritten as
B 22ϕ 1 Bϕ B 22 ϕ
∆ϕ “  2 ` 1  `  2 “ 0
 = Br 2  r Br  Bz2  0
r
r r
(2)
z
(2)
The boundary
conditions of the Laplace equation yield the following conclusions:
The boundary conditions of the Laplace equation yield the following conclusions:
(A) At the
betweenbetween
electrodes
and medium,
the potential
difference
between
the inner
(A)interface
At the interface
electrodes
and medium,
the potential
difference
between
the and
inner and
outer annular
electrode
is expressed
as ΔV, i.e.,
outer annular
electrode
is expressed
as ∆V,
i.e.,

 V “ ∆V
ϕ|z“z 0,r
´ zϕ|
0, rii ă
 rră
r rio
 0,zr“0,r
r  roi ărăroo
ii
io
oi
(3)
oo
(3)
(B) In other areas of this interface, the z-direction component of the electric field intensity is zero.
(B) In other areas of this interface, the z-direction component of the electric field intensity is zero.
ˇ
ˇ
ˇ
Bϕ ˇˇ
Bϕ ˇˇ
Bϕ ˇˇ
“ 
“
“0


Bz ˇz“0,0ărării = Bz ˇz“0,rio ărăroi
Bz ˇz“0,roo
ă0răr8
z
z  0,0  r  rii
z
z  0, rio  r  roi
z
(4)
(4)
z  0, roo  r  r
(C) At the bottom surface of the medium, the z-direction component of the electric field intensity
is zero.
(C) At the bottom surface of the medium, ˇˇ the z-direction component of the electric field
Bϕ ˇ
“0
(5)
intensity is zero.
Bz ˇz“h
(D) Through sector integration of the electric
 density, the approximate equations of the electric
0
(5)
quantity on the inner and the outer annular
can be expressed as
z electrodes
z h
ż θ0 ż rio ˆ
ˇ
˙
2
2
˜
ˇ
¸
Bϕ ˇ
r ´ rii
Bϕ ˇˇ
(D) Through sector integration
of the electric
Q«
ε ´ ofˇˇ the electric
rdrdθdensity,
« ε io the approximate
θ0 ´ equations
ˇ
Bz
2
Bz
quantity on0the rinner
and
the
outer
annular
electrodes
can
be
expressed
as
z“0
z“0,rii ărărio
ii
(6)
˜
¸
ˇ ˙
ˇ
ż θ0 ż roo ˆ
Bϕ ˇˇ
roo 2 ´ roi 2
Bϕ ˇˇ
´Q «
ε ´
rdrdθ « ε
θ0 ´ ˇ
Bz ˇz“0
2
Bz z“0,roi ărăroo
0
roi
(7)
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Subsequently, we obtain another two conditions, i.e.,
´
ˇ
2Q
Bϕ ˇˇ
˘
« ` 2
Bz ˇz“0,rii ărărio
ε rio ´ rii 2 θ0
(8)
ˇ
2Q
Bϕ ˇˇ
˘
«´ ` 2
´ ˇ
Bz z“0,roi ărăroo
ε roo ´ roi 2 θ0
(9)
To solve the Laplace equation, a zeroth-order Hankel transform is employed, and Equation (2) is
expressed as
B 2 Ψ pξ, zq
´ ξ 2 Ψ pξ, zq “ 0
(10)
Bz2
A general solution is easily obtained as
Ψ pξ, zq “ K1 e´ξz ` K2 eξz
(11)
For boundary condition C, we have
ˇ
BΨ pξ, zq ˇˇ
“0
Bz ˇz“h
(12)
For boundary conditions B and D, we have
ff
«
ˇ
BΨ pξ, zq ˇˇ
2Q
rio J1 pξrio q ´ rii J1 pξrii q roo J1 pξroo q ´ roi J1 pξroi q
˘
`
˘
`
´
“
¨
´
Bz ˇz“0
εθ0
ξ r2 io ´ r2 ii
ξ r2 oo ´ r2 oi
(13)
By solving Equation (11) using transformed boundary Equations (12) and (13), we obtain
a particular solution for Equation (10).
«
ff
2Q cosh rξ ph ´ zqs
rio J1 pξrio q ´ rii J1 pξrii q roo J1 pξroo q ´ roi J1 pξroi q
˘
`
˘
`
Ψ pξ, zq “
¨
´
εθ0 ξsinhpξhq
ξ r2 io ´ r2 ii
ξ r2 oo ´ r2 oi
(14)
Therefore, the inverse zeroth-order Hankel transform of Equation (14) helps solve the initial
Laplace equation.
ż8
ϕ pr, zq “
0
Ψ pξ, zqJ0 pξrq ξdξ
(15)
We utilize the average value over the surface integration as an approximation of the electrode
electric potential, i.e.,
şr
θ0 riiio ϕ pr, zq|z“0 rdr
ϕ|z“0,rii ărărio “
(16)
˘
θ0 ` 2
r io ´ r2 ii
2
şroo
θ0 roi ϕ pr, zq|z“0 rdr
ϕ|z“0,roi ărăroo “
(17)
˘
θ0 ` 2
r oo ´ r2 oi
2
By combining the boundary condition in A in Equation (3), we obtain
«
ż
4Q 8
∆V “
εθ0 0
rio J1 pξrio q ´ rii J1 pξrii q roo J1 pξroo q ´ roi J1 pξroi q
`
˘
`
˘
´
ξ r2 io ´ r2 ii
ξ r2 oo ´ r2 oi
tanhpξhq
ff2
dξ
(18)
2
 r J  r   r J  r  r J  r   r J  r  
ii 1
ii
oi 1
oi
 oo 1 oo2
 io 1 io2

2
2
  r io  r ii 
  r oo  r oi 

4Q  
V 
d

0
 0
tanh( h)
(18)
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According to the definition of capacitance, an analytical expression of capacitance value C can
be obtained as
According to the definition of capacitance, an analytical expression of capacitance value C can be
obtained as
Q
1
C
C“
  0 
QV
“ ε ¨ θ0 ¨
∆V
2
1
« rio J1  rio   rii J1  rii  roo J1  roo   roi J1  roi ff 2
 rio J1 pξrio2q ´ rii J21 pξrii q roo J1 pξroo 2q ´ roi J21 pξroi q 
 ` r io  r ii˘
` r oo  r oi˘
´

 
ξ r2 io ´ r2 ii
ξ r2 oo ´ r2 oi
4ş8 
d
4 00
dξ
tanh( h)
tanhpξhq




(19)
(19)

To
validate the
theanalytical
analyticalmodel
modelinin
Equation
(19),
capacitance
value
is also
calculated
by a
To validate
Equation
(19),
capacitance
value
C is Calso
calculated
by a finite
finite
element
method.
The electrostatic
field analysis
onelement
a finitemethod
elementismethod
available
element
method.
The electrostatic
field analysis
based onbased
a finite
availableisin
ANSYS,
in
ANSYS,
a
finite
element
program.
Firstly,
we
concentrate
on
the
parameters
of
the
medium,
a finite element program. Firstly, we concentrate on the parameters of the medium, permittivity ε
permittivity
ε and
thickness
h. Figure
shows that
capacitance
value
C is strictlytoproportional
and thickness
h. Figure
3a shows
that 3a
capacitance
value
C is strictly
proportional
permittivitytoε,
permittivity
ε, where
the from
resultfinite
obtained
from
finite agrees
element
method
agrees
well
with that
of the
where the result
obtained
element
method
well
with that
of the
proposed
analytical
˝
proposed
analytical
model.
The
default
settings
of
other
parameters
are
listed
as
follows:
h
=
10
model. The default settings of other parameters are listed as follows: h = 10 mm, θ 0 = 60 , rii = 7mm,
mm,
θrio
0 ==60°,
rii = r7oimm,
= 9 and
mm,roo
roi =
= 9.5
and
roo = 11.5
mm. Figure
3b shows that
9 mm,
= 9.5riomm
11.5mm
mm.
Figure
3b shows
that capacitance
valuecapacitance
C increasesvalue
with
C
increases
with medium
h when medium
is 1 × ε0. When
h increases
beyond
medium
thickness
h whenthickness
medium permittivity
ε is 1 permittivity
ˆ ε0 . When hε increases
beyond
a critical value
of
aapproximately
critical value of
approximately
6
mm,
the
ascending
trend
disappears.
6 mm, the ascending trend disappears.
0.8
(b)
Analytical Model
Finite Element Method
0.7
Capacitance value, C (pF)
Capacitance value, C (pF)
(a)
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.10
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0
2
4
6
8
10
Medium permittivity, ε (×ε0)
Analytical Model
Finite Element Method
0.09
0
2
4
6
8
10
12
14
Medium thickness, h (mm)
16
18
Figure
3. Capacitance
Capacitancevalues
valuesof of
capacitors
(a) different
medium
permittivity
ε different
and (b)
Figure 3.
thethe
capacitors
withwith
(a) different
medium
permittivity
ε and (b)
different
medium
thickness
h.
medium thickness h.
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The next analysis is conducted under the condition
condition that
that medium
medium permittivity
permittivity εεisisfixed
fixedatat11ˆ× εε00
and thickness is fixed at 15 mm,
mm, where
where εε00 is vacuum
vacuum permittivity.
permittivity. We subsequently consider the
capacitor
electrodes.
Different
values
of central
angle θ 0angle
are setθfor
electrodes,
geometric parameters
parametersofof
capacitor
electrodes.
Different
values
of central
0 are
set for
and both calculation
and simulation
are shown
Figure
4. in
The
capacitance
value C is strictly
electrodes,
and both calculation
and results
simulation
resultsin
are
shown
Figure
4. The capacitance
value
proportional
to
central
angle
θ
of
the
electrodes.
C is strictly proportional to central
angle θ0 of the electrodes.
0
Capacitance value, C (pF)
0.5
Analytical Model
Finite Element Method
0.4
0.3
0.2
0.1
0.0
0
60
120
180
240
Central angle, θ0 (°)
300
360
Figure
central angle
angle θθ0..
Figure 4.
4. Capacitance
Capacitance values
values of
of the
the capacitors
capacitors with
with different
different central
0
Finally, four radial parameters of capacitor electrodes, namely, rii, rio, roi, and roo, are
individually studied while central angle θ0 is fixed as a default setting of 60°. Figure 5 shows the
results. In general, C increases with the geometric size of capacitor electrodes. When rii increases
from 5.0 mm to 7.0 mm, as shown in Figure 5a, C remains almost unchanged. The same result can be
easily observed in Figure 5d when roo is in the interval from 11.5 mm to 13.5 mm. This result
Cap
0.1
0.0
0
60
120
180
240
300
Central angle, θ0 (°)
360
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Figure 4. Capacitance values of the capacitors with different central angle θ0.
Finally,
fourradial
radial
parameters
of capacitor
electrodes,
, rior, oor,oiare
, and
roo, are
Finally, four
parameters
of capacitor
electrodes,
namely,namely,
rii , rio , roir, iiand
individually
˝
individually
angle θ0 is fixed as a default setting of 60°. Figure 5 shows the
studied whilestudied
central while
angle θcentral
0 is fixed as a default setting of 60 . Figure 5 shows the results. In general,
results.
In
general,
C
increases
with
geometric
size ofWhen
capacitor
electrodes.
rii to
increases
C increases with the geometric size of the
capacitor
electrodes.
rii increases
fromWhen
5.0 mm
7.0 mm,
from
5.0
mm
to
7.0
mm,
as
shown
in
Figure
5a,
C
remains
almost
unchanged.
The
same
result
can be
as shown in Figure 5a, C remains almost unchanged. The same result can be easily observed
in
easily
when from
roo is 11.5
in the
from This
11.5 result
mm to
13.5 mm.
Figureobserved
5d when in
roo Figure
is in the5dinterval
mminterval
to 13.5 mm.
indicates
thatThis
for aresult
fixed
indicates
that for two
a fixed
distancethe
between
two
electrodes,
the electrode
size’s in
contribution
the
distance between
electrodes,
electrode
size’s
contribution
to the increase
capacitancetovalue
increase
in capacitance
becomes
increasingly
lesser
when radial
reaches a2certain
C becomes
increasinglyvalue
lesserCwhen
radial
width reaches
a certain
extent.width
For example,
mm is
extent.
For
example,
2
mm
is
a
radial
width
limit
for
electrodes
in
the
proposed
model
when
the
a radial width limit for electrodes in the proposed model when the distance between two electrodes
is
distance
between
two
electrodes
is
approximately
0.5
mm.
approximately 0.5 mm.
0.10
Analytical Model
Finite Element Method
Capacitance value, C (pF)
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
(b)
5.5
6.0
6.5
7.0
7.5
8.0
8.5
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.00
7.0
9.0
Inner radius of the inner annular electrode, rii (mm)
Analytical Model
Finite Element Method
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
7.5
8.0
8.5
9.0
9.5
Outer radius of the inner annular electrode, rio (mm)
(d)
0.10
Analytical Model
Finite Element Method
0.09
Capacitance value, C (pF)
0.11
0.10
Capacitance value, C (pF)
Analytical Model
Finite Element Method
0.01
0.00
5.0
(c)
0.11
0.10
Capacitance value, C (pF)
(a)
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
9.0
9.5
10.0
10.5
11.0
11.5
Inner radius of the outer annular electrode, roi (mm)
0.00
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
Outer radius of the outer annular electrode, roo (mm)
Figure
Figure 5.
5. Capacitance
Capacitancevalues
valuesofofthe
thecapacitors
capacitorswith
withdifferent
different parameters
parametersin
in the
the radial
radial direction:
direction:
(a)
9 mm,
roi =roi9.5=mm,
and rand
oo = 11.5
(b)mm;
rii = 7(b)
mm,
mm,
roomm,
= 11.5and
mm,
(c)=rii11.5
= 7 mm,
(a)rrioio= =
9 mm,
9.5 mm,
roo mm,
= 11.5
rii r=oi =9.5
7 mm,
roiand
=9.5
roo
mm;
io =r9 mm,
and
r
oo=
=
11.5
mm,
and
(d)
r
ii =mm;
7
mm,
r
io(d)
=
9
mm,
and
r
oi
=
9.5
mm.
r(c)
=
7
mm,
r
9
mm,
and
r
=
11.5
and
r
=
7
mm,
r
=
9
mm,
and
r
=
9.5
mm.
oo
ii
io
ii
io
oi
The overall trend of two curves shown in Figure 5 agrees well, which indicates that the proposed
analytical model is a suitable approximation of a real capacitor. Let us consider a situation in which the
distance between two electrodes remains the same, whereas the radial length of electrodes becomes
smaller, namely, rii approaches rio = 9 mm, as shown in Figure 5a, and roo approaches roi = 9.5 mm, as
shown in Figure 5d. We notice that the difference between analytical model and result from the finite
element method becomes larger. It is because the fringing effect between two electrodes in the real
model is negligible at this time, and the proposed analytical model does not apply any more. Further,
when the distance between two annular electrodes becomes larger than 1 mm, namely, rio approaches
rii = 7 mm (Figure 5b) and roi approaches roo = 11.5 mm (Figure 5c), the proposed analytical model is
not sufficiently accurate. It is because the normal capacitance is dominant in the capacitance value C in
this condition.
Table 1 shows a concise conclusion on how different parameters affect capacitance value C. Based
on the analysis result mentioned above, we optimize the structure parameters of a tilt sensor that uses
segmented annular coplanar capacitors shown in Figure 1c,d.
(Figure 5c), the proposed analytical model is not sufficiently accurate. It is because the normal
capacitance is dominant in the capacitance value C in this condition.
Table 1 shows a concise conclusion on how different parameters affect capacitance value C.
Based on the analysis result mentioned above, we optimize the structure parameters of a tilt sensor
that uses segmented annular coplanar capacitors shown in Figure 1c,d.
Sensors 2016, 16, 133
Table 1. Contributions of different control parameters to capacitance value C.
Control Parameters
Contribution to C
Central angle θ0
Strictly proportional to C; designed to satisfy various needs
7 of 11
Table 1. Contributions of different control parameters to capacitance value C.
Contribution to C
Control Parameters
Medium
Central angle
θ 0 thickness h
Supposed
to be proportional
large enough to guarantee
a large
C various needs
Strictly
C; designed
to satisfy
Medium Inner
thickness
h of inner annular
radius
Supposed
be large
enough atolarge
guarantee
a large C
Not necessarily
tooto
small
to guarantee
C; designed
according
to necessarily
the value oftoo
rio small to guarantee a large C; designed
Not
electrode rii
Inner radius of inner annular electrode rii
according to the value of rio
Outer radius of inner annular
Determining
the between
distance two
between
two electrodes
and
Outer radius
of inner
rio
electrode
rio annular electrodeDetermining
the distance
electrodes
and
assumed
to be
smalltoenough
to guarantee
Inner radius
outer of
annular
electrode assumed
roi
to be small
enough
guarantee
a large C a large C
Innerofradius
outer annular
electrode roi
Outer radius of outer annular electrode roo
Outer radius of outer annular
electrode roo
Not necessarily too large to guarantee a large C; designed
according
the to
value
of roi a large C; designed
Not necessarily
tooto
large
guarantee
according to the value of roi
3. Implementation and Experiment
3. Implementation and Experiment
According
to the analytical result, capacitance value C of a segmented annular coplanar capacitor
According to the analytical result, capacitance value C of a segmented annular coplanar
is linearly
proportional
toproportional
medium permittivity
ε as well as
central
θ 0angle
. From
tilt sensor
capacitor
is linearly
to medium permittivity
ε as
well asangle
central
θ0. this
Fromfact,
this a
fact,
is designed,
and its
sensing mechanism
is mechanism
shown in is
Figure
a tilt sensor
is designed,
and its sensing
shown6.in Figure 6.
Figure
6. Schematic
a tilt
sensorwith
withfour
four segmented
concentric
annular
coplanar
capacitors.
Figure 6.
Schematic
viewview
of aoftilt
sensor
segmented
concentric
annular
coplanar
capacitors.
Four segmented annular coplanar capacitors with central angle θ 0 = 88˝ mentioned above are
centrally symmetrically distributed on a dielectric substrate, as shown in Figure 6. These four capacitors
are half-immersed in non-conducting liquid of which the level crosses the common center of the
electrodes. Each segmented capacitor consists of two parts: the capacitance in the substrate side and
the capacitance at the opposite side. We designate the thicknesses of the substrate and the liquid as
hsub and hliq , respectively.
The tilt sensor rotates clockwise (or anti-clockwise) with a small inclination angle α, whereas
the liquid remains relatively static. Consequently, the capacitance values of the left and right
capacitors change with the rotation. Four capacitance values in the tilt sensor can be described
by the following equations:
22π
22π
¨ ε sub ` Kliq ¨
¨ ε liq
45
45
ˆ
˙
ˆ
˙
22π
11π
11π
“ Ksub ¨
¨ ε sub ` Kliq ¨
` α ¨ ε air ` Kliq ¨
´ α ¨ ε liq
45
45
45
Cdown “ Ksub ¨
Cleft
22π
22π
¨ ε sub ` Kliq ¨
¨ ε air
45
45
ˆ
˙
ˆ
˙
22π
11π
11π
“ Ksub ¨
¨ ε sub ` Kliq ¨
´ α ¨ ε air ` Kliq ¨
` α ¨ ε liq
45
45
45
Cup “ Ksub ¨
Cright
(20)
(21)
(22)
(23)
Sensors 2016, 16, 133
8 of 11
where the ratio parameters Ksub and Kliq satisfy the following equations:
1
Ksub “
«
4
ş8
rio J1 pξrio q ´ rii J1 pξrii q roo J1 pξroo q ´ roo J1 pξroi q
˘
˘
`
`
´
ξ r2 io ´ r2 ii
ξ r2 oo ´ r2 oi
0
(24)
ff2
tanhpξhsub q
dξ
1
Kliq “
«
ş8
40
rio J1 pξrio q ´ rii J1 pξrii q roo J1 pξroo q ´ roo J1 pξroi q
˘
˘
`
`
´
ξ r2 io ´ r2 ii
ξ r2 oo ´ r2 oi
tanhpξhliq q
(25)
ff2
dξ
Through simple derivation from Equations (20)–(24), we obtain the equation for α.
α“
11π Cright ´ Cleft
¨
45 Cdown ´ Cup
(26)
We should note that Cup and Cdown are constant when the tilt angle is within ˘44˝ because the
corresponding electrodes are fully exposed in air or immersed in liquid. Further, under this condition,
the sensitivity of this tilt sensor can be analyzed as expressed in the following equation:
´
¯
Cright ´ Cleft “ 2 ¨ Kliq ¨ ε liq ´ ε air ¨ α “ Cdif ¨ α
(27)
We introduce parameter Cdif to simplify the analysis process. In Equation (27), Cdif represents the
capacitance value of a segmented annular coplanar capacitor whose parameters are listed as follows:
θ 0 = 2 rad, ε = εliq ´εair , and h = hliq . Radial sizes rii , rio , roi , and roo are the same as those of the
capacitors in this study. Theoretically speaking, a larger Cdif results in better sensitivity of the proposed
tilt sensor.
According to the analysis presented in Section 2, a smaller distance between two electrodes
can easily lead to a large Cdif . Because of the limit in the conventional technology, in most
cases, radial distance D = roi ´rio , cannot be fabricated at a small-scale level. Here, we utilize
the printed-circuit-board technology to fabricate the sensor, where a 0.2-mm-wide distance can be
accurately ensured. Under this condition, the inner and outer boundaries of electrodes are set as
rii = 7 mm and roo = 11.5 mm after an overall consideration because 4.5 mm is sufficiently wide
to maintain a large capacitance value. In reality, the permittivity of air εair is constant, and the
non-conducting liquid with a larger permittivity εliq is preferred where glycerol (εliq « 42.5 ˆ ε0 ) is
employed. By considering liquid thickness hliq , 15 mm is sufficient.
Under these conditions, we optimize the values of rio and roi , namely, the position of the gap
rg = (rio + roi )/2, using the proposed analytical model as well as the finite element method. Figure 7
shows that when the gap moves from the inner to the outer boundary, Cdif first increases and then
decreases, reaching a peak of 7.382 pF when rg is equal to 10 mm. Finally, we choose rio = 9.9 mm and
roi = 10.1 mm to realize the best sensitivity of the tilt sensor.
To verify the accuracy of the analysis process, three models with different gap position (rg is equal
to 9 mm, 10 mm, and 11 mm for Models 1, 2, and 3, respectively) are fabricated utilizing printed circuit
board (PCB) technology, as shown in Figure 8a. The electrodes of the planar capacitors are made of
metal copper covered with the tin which prevent the oxidation process of metal copper. The dielectric
substrate is made of fiberglass resin, which is a commonly used material for PCB. Limited by the
conventional technology, the electrode thickness is about 0.035 mm. Eight holes on each model are
utilized for precise measurement of the capacitance value. Measurement tests on the relationship
between Cright ´ Cleft and α are conducted on a standard tilt platform, and Figure 8b shows the
percentage change of Cright and Cleft with respect to the inclination change in three models. In Figure 8c,
distance D = roi−rio, cannot be fabricated at a small-scale level. Here, we utilize the
printed-circuit-board technology to fabricate the sensor, where a 0.2-mm-wide distance can be
accurately ensured. Under this condition, the inner and outer boundaries of electrodes are set as rii = 7
mm and roo = 11.5 mm after an overall consideration because 4.5 mm is sufficiently wide to maintain a
large capacitance value. In reality, the permittivity of air εair is constant, and the non-conducting
Sensors liquid
2016, 16,with
133 a larger permittivity εliq is preferred where glycerol (εliq ≈ 42.5 × ε0 ) is employed. By 9 of 11
considering liquid thickness hliq, 15 mm is sufficient.
Under these conditions, we optimize the values of rio and roi, namely, the position of the gap
Cright ´
and
α accurately
dovetailanalytical
with a linear
derived
from Equation
(27). Model
2
left
rg =C(r
io +
roi)/2,
using the proposed
modelrelationship
as well as the
finite element
method. Figure
7
yieldsshows
the highest
sensitivity
0.129from
pF/˝the
, followed
by Model
1 (0.120 pF/
) and
Model and
3 (0.109
that when
the gapof
moves
inner to the
outer boundary,
Cdif ˝first
increases
then pF/˝ ).
decreases, reaching
a peakof
ofthree
7.382 pF
when is
rg calculated
is equal to 10
mm.
Finally,
io = 9.9
and
Corresponding
non-linearity
models
and
found
to we
be choose
below r0.5%
FSmm
(full
scale of
˝
r
oi = 10.1 mm to realize the best sensitivity of the tilt sensor.
˘40 ) for Models 1 and 3 and below 0.4% FS for Model 2.
Analytical Model
Finite Element Method
Capacitance value, Cdif (pF)
8
7
6
5
4
3
2
1
0
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Position of the gap, rg (mm)
10.5
11.0
11.5
7. Capacitance
values
withdifferent
different gap
gap positions
ii = 7
roo =roo
11.5
mm, mm,
θ0 = 2θrad,
FigureFigure
7. Capacitance
values
with
positions(r(r
7 mm,
= 11.5
ii = mm,
0 = 2 rad,
= 15 mm,
ε = 41.5
h = 15hmm,
and and
ε = 41.5
ˆ ε×0ε).0).
Sensors 2016, 16, 133
10 of 12
To verify the accuracy of the analysis process, three models with different gap position (rg is
equal to 9 mm, 10 mm, and 11 mm for Models 1, 2, and 3, respectively) are fabricated utilizing
printed circuit board (PCB) technology, as shown in Figure 8a. The electrodes of the planar
capacitors are made of metal copper covered with the tin which prevent the oxidation process of
metal copper. The dielectric substrate is made of fiberglass resin, which is a commonly used material
for PCB. Limited by the conventional technology, the electrode thickness is about 0.035 mm. Eight
holes on each model are utilized for precise measurement of the capacitance value. Measurement
tests on the relationship between Cright − Cleft and α are conducted on a standard tilt platform, and
Figure 8b shows the percentage change of Cright and Cleft with respect to the inclination change in
three models. In Figure 8c, Cright − Cleft and α accurately dovetail with a linear relationship derived
from Equation (27). Model 2 yields the highest sensitivity of 0.129 pF/°, followed by Model 1 (0.120
pF/°) and Model 3 (0.109 pF/°). Corresponding non-linearity of three models is calculated and found
to be below 0.5% FS (full scale of ±40°) for Models 1 and 3 and below 0.4% FS for Model 2.
Figure
8. (a)8.three
models
with
different
(b)percentage
percentagechange
change
Cright
Figure
(a) three
models
with
differentstructure
structureparameters;
parameters; (b)
of of
Cright
and and
Cleft Cleft
respect
to inclination
the inclination
change
threemodels;
models; (c) corresponding
comparison
results.
respect
to the
change
ininthree
correspondingsensitivity
sensitivity
comparison
results.
In addition, the accuracy test of Model 2 is also performed on the standard tilt platform. When
In addition, the accuracy test of Model 2 is also performed on the standard tilt platform. When the
the inclination angle of the platform varies˝by 5° per step from −40°
to 40°, the capacitance values of
inclination
angle of the platform varies by 5 per step from ´40˝ to 40˝ , the capacitance values of four
four segmented coplanar capacitors are recorded, and corresponding α is calculated using Equation
segmented
coplanar
capacitors
are recorded,
is calculated
usingthat
Equation
(26). The
results for
10 measurement
timesand
are corresponding
shown in Figureα 9,
which indicates
a 0.4° (26).
˝ accuracy
The results
for
10
measurement
times
are
shown
in
Figure
9,
which
indicates
that
a
0.4
accuracy is achieved. It should be mentioned that the highest measurement errors are found for
is achieved.
It should be
mentioned
that the
highest
errors are
found for negative
negative inclination
angles.
One possible
reason
mightmeasurement
be the manufacturing
non-uniformity
in
electrode
gap. When
Model 2 rotates
negative inclination
angles), in
theelectrode
left
inclination
angles.
One possible
reason anti-clockwise
might be the (with
manufacturing
non-uniformity
electrodes get into liquid gradually and the increase of Cleft is not strictly proportional to the
inclination change. The non-uniformity in the left electrode gap might cause a higher measurement
error for negative inclination angles.
0.8
0.7
respect to the inclination change in three models; (c) corresponding sensitivity comparison results.
Measurement error (°)
In addition, the accuracy test of Model 2 is also performed on the standard tilt platform. When
the inclination angle of the platform varies by 5° per step from −40° to 40°, the capacitance values of
four segmented coplanar capacitors are recorded, and corresponding α is calculated using Equation
Sensors(26).
2016,The
16, 133
10 of 11
results for 10 measurement times are shown in Figure 9, which indicates that a 0.4°
accuracy is achieved. It should be mentioned that the highest measurement errors are found for
negative inclination angles. One possible reason might be the manufacturing non-uniformity in
gap. When
Model
rotates
anti-clockwise
(with negative
inclination
angles), theangles),
left electrodes
electrode
gap. 2When
Model
2 rotates anti-clockwise
(with
negative inclination
the left get
into liquid
gradually
the gradually
increase of
Cleft
not strictly
proportional
to proportional
the inclination
change.
electrodes
get intoand
liquid
and
theisincrease
of Cleft
is not strictly
to the
inclination
change.
The
non-uniformity
in
the
left
electrode
gap
might
cause
a
higher
measurement
The non-uniformity in the left electrode gap might cause a higher measurement error for negative
error for
negative inclination angles.
inclination
angles.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-45 -40 -35 -30 -25 -20 -15 -10 -5
0
5 10 15 20 25 30 35 40 45
Inclination angle, α (°)
Figure
9. Accuracy
tiltsensor
sensorwith
with
Model
Figure
9. Accuracytest
testof
ofthe
the proposed
proposed tilt
Model
2. 2.
We chose 88˝ as the central angle of fabricated devices because of a tradeoff between the accuracy
and the measurement range of the proposed tilt sensor. The central angle of 88˝ means that the
segmenting angle between two adjacent inner annular electrodes is 2˝ . In terms of a central angle larger
than 88˝ , adjacent annular electrodes are too close to each other that the cross capacitance between two
coplanar capacitors might influence the accuracy of the final measurement result. In terms of a central
angle smaller than 88˝ , the segmenting angle, as well as the insensitive zone of the tilt sensor, becomes
larger. Then, the measurement range decreases as a result. However, such a design is not unique and
other parameters for coplanar capacitors could be tried in future work.
4. Conclusions
An investigation on a segmented annular coplanar capacitor has been presented, and its analytical
model has been established. By solving a Laplace equation with Hankel transform, a mathematical
expression of the capacitance value is derived. The finite element method verifies the analytical result
well. The dimension parameters of the coplanar capacitor are individually studied, and we obtain
a general principle on their contributions to the capacitance value. Consequently, we analyze and
optimize the structure parameters of a segmented coplanar capacitive tilt sensor utilizing the proposed
analytical model. Three models with different positions of the electrode gap are fabricated and tested.
The experiment results show that Model 2 (rg = 10 mm) yields a high sensitivity: 0.129 pF/˝ with
a non-linearity of <0.4% FS and an accuracy of 0.4˝ is achieved. When the total width for two electrodes
is fixed, the width of the inner annular electrode should be larger than the width of the outer annular
electrode to realize the best solution. The optimal width ratio Kwid is related with both inner radii of
inner annular electrode rii and outer radii of outer annular electrode roo . This finding offers plenty of
opportunities for various measurement requirements in addition to achieving an optimized structure
in practical design.
Acknowledgments: This work was financially supported by the key support program of the major research
project of China National Natural Science Foundation (Grant No. 91536224) and the Fundamental Research Funds
for the Central Universities (Grant No. HIT.NSRIF.20168).
Author Contributions: The work presented here was carried out in collaboration between all authors. Jiahao Guo
designed the new idea and wrote the manuscript. Pengcheng Hu was involved in the mathematical development
and carried out the experiments. Jiubin Tan drew the main conclusions and critically reviewed the paper.
Sensors 2016, 16, 133
11 of 11
Conflicts of Interest: The authors declare no conflict of interest.
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